def _element_constructor_(self, x, n=0):
r"""
Construct a Laurent series from `x`.
INPUT:
- ``x`` -- object that can be converted into a Laurent series
- ``n`` -- (default: 0) multiply the result by `t^n`
EXAMPLES::
sage: R.<u> = LaurentSeriesRing(Qp(5, 10))
sage: S.<t> = LaurentSeriesRing(RationalField())
sage: print R(t + t^2 + O(t^3))
(1 + O(5^10))*u + (1 + O(5^10))*u^2 + O(u^3)
Note that coercing an element into its own parent just produces
that element again (since Laurent series are immutable)::
sage: u is R(u)
True
Rational functions are accepted::
sage: I = sqrt(-1)
sage: K.<I> = QQ[I]
sage: P.<t> = PolynomialRing(K)
sage: L.<u> = LaurentSeriesRing(QQ[I])
sage: L((t*I)/(t^3+I*2*t))
1/2 + 1/4*I*u^2 - 1/8*u^4 - 1/16*I*u^6 + 1/32*u^8 +
1/64*I*u^10 - 1/128*u^12 - 1/256*I*u^14 + 1/512*u^16 +
1/1024*I*u^18 + O(u^20)
::
sage: L(t*I) / L(t^3+I*2*t)
1/2 + 1/4*I*u^2 - 1/8*u^4 - 1/16*I*u^6 + 1/32*u^8 +
1/64*I*u^10 - 1/128*u^12 - 1/256*I*u^14 + 1/512*u^16 +
1/1024*I*u^18 + O(u^20)
TESTS:
When converting from `R((z))` to `R((z))((w))`, the variable
`z` is sent to `z` rather than to `w` (see :trac:`7085`)::
sage: A.<z> = LaurentSeriesRing(QQ)
sage: B.<w> = LaurentSeriesRing(A)
sage: B(z)
z
sage: z/w
z*w^-1
Various conversions from PARI (see also :trac:`2508`)::
sage: L.<q> = LaurentSeriesRing(QQ)
sage: L.set_default_prec(10)
doctest:...: DeprecationWarning: This method is deprecated.
See http://trac.sagemath.org/16201 for details.
sage: L(pari('1/x'))
q^-1
sage: L(pari('polchebyshev(5)'))
5*q - 20*q^3 + 16*q^5
sage: L(pari('polchebyshev(5) - 1/x^4'))
-q^-4 + 5*q - 20*q^3 + 16*q^5
sage: L(pari('1/polchebyshev(5)'))
1/5*q^-1 + 4/5*q + 64/25*q^3 + 192/25*q^5 + 2816/125*q^7 + O(q^9)
sage: L(pari('polchebyshev(5) + O(x^40)'))
5*q - 20*q^3 + 16*q^5 + O(q^40)
sage: L(pari('polchebyshev(5) - 1/x^4 + O(x^40)'))
-q^-4 + 5*q - 20*q^3 + 16*q^5 + O(q^40)
sage: L(pari('1/polchebyshev(5) + O(x^10)'))
1/5*q^-1 + 4/5*q + 64/25*q^3 + 192/25*q^5 + 2816/125*q^7 + 8192/125*q^9 + O(q^10)
sage: L(pari('1/polchebyshev(5) + O(x^10)'), -10) # Multiply by q^-10
1/5*q^-11 + 4/5*q^-9 + 64/25*q^-7 + 192/25*q^-5 + 2816/125*q^-3 + 8192/125*q^-1 + O(1)
sage: L(pari('O(x^-10)'))
O(q^-10)
"""
from sage.rings.fraction_field_element import is_FractionFieldElement
from sage.rings.polynomial.polynomial_element import is_Polynomial
from sage.rings.polynomial.multi_polynomial_element import is_MPolynomial
from sage.structure.element import parent
P = parent(x)
if isinstance(x, self.element_class) and n==0 and P is self:
return x # ok, since Laurent series are immutable (no need to make a copy)
elif P is self.base_ring():
# Convert x into a power series; if P is itself a Laurent
# series ring A((t)), this prevents the implementation of
# LaurentSeries.__init__() from effectively applying the
# ring homomorphism A((t)) -> A((t))((u)) sending t to u
# instead of the one sending t to t. We cannot easily
# tell LaurentSeries.__init__() to be more strict, because
# A((t)) -> B((u)) is expected to send t to u if A admits
# a coercion to B but A((t)) does not, and this condition
# would be inefficient to check there.
x = self.power_series_ring()(x)
elif isinstance(x, pari_gen):
t = x.type()
if t == "t_RFRAC": # Rational function
x = self(self.polynomial_ring()(x.numerator())) / \
self(self.polynomial_ring()(x.denominator()))
return (x << n)
elif t == "t_SER": # Laurent series
n += x._valp()
bigoh = n + x.length()
x = self(self.polynomial_ring()(x.Vec()))
return (x << n).add_bigoh(bigoh)
else: # General case, pretend to be a polynomial
return self(self.polynomial_ring()(x)) << n
elif is_FractionFieldElement(x) and \
(x.base_ring() is self.base_ring() or x.base_ring() == self.base_ring()) and \
(is_Polynomial(x.numerator()) or is_MPolynomial(x.numerator())):
x = self(x.numerator())/self(x.denominator())
return (x << n)
return self.element_class(self, x, n)
def __call__(self, x, n=0):
r"""
Coerces the element x into this Laurent series ring.
INPUT:
- ``x`` - the element to coerce
- ``n`` - the result of the coercion will be
multiplied by `t^n` (default: 0)
EXAMPLES::
sage: R.<u> = LaurentSeriesRing(Qp(5, 10))
sage: S.<t> = LaurentSeriesRing(RationalField())
sage: print R(t + t^2 + O(t^3))
(1 + O(5^10))*u + (1 + O(5^10))*u^2 + O(u^3)
Note that coercing an element into its own parent just produces
that element again (since Laurent series are immutable)::
sage: u is R(u)
True
Rational functions are accepted::
sage: I = sqrt(-1)
sage: K.<I> = QQ[I]
sage: P.<t> = PolynomialRing(K)
sage: L.<u> = LaurentSeriesRing(QQ[I])
sage: L((t*I)/(t^3+I*2*t))
1/2 + 1/4*I*u^2 - 1/8*u^4 - 1/16*I*u^6 + 1/32*u^8 +
1/64*I*u^10 - 1/128*u^12 - 1/256*I*u^14 + 1/512*u^16 +
1/1024*I*u^18 + O(u^20)
::
sage: L(t*I) / L(t^3+I*2*t)
1/2 + 1/4*I*u^2 - 1/8*u^4 - 1/16*I*u^6 + 1/32*u^8 +
1/64*I*u^10 - 1/128*u^12 - 1/256*I*u^14 + 1/512*u^16 +
1/1024*I*u^18 + O(u^20)
Various conversions from PARI (see also #2508)::
sage: L.<q> = LaurentSeriesRing(QQ)
sage: L.set_default_prec(10)
sage: L(pari('1/x'))
q^-1
sage: L(pari('poltchebi(5)'))
5*q - 20*q^3 + 16*q^5
sage: L(pari('poltchebi(5) - 1/x^4'))
-q^-4 + 5*q - 20*q^3 + 16*q^5
sage: L(pari('1/poltchebi(5)'))
1/5*q^-1 + 4/5*q + 64/25*q^3 + 192/25*q^5 + 2816/125*q^7 + O(q^9)
sage: L(pari('poltchebi(5) + O(x^40)'))
5*q - 20*q^3 + 16*q^5 + O(q^40)
sage: L(pari('poltchebi(5) - 1/x^4 + O(x^40)'))
-q^-4 + 5*q - 20*q^3 + 16*q^5 + O(q^40)
sage: L(pari('1/poltchebi(5) + O(x^10)'))
1/5*q^-1 + 4/5*q + 64/25*q^3 + 192/25*q^5 + 2816/125*q^7 + 8192/125*q^9 + O(q^10)
sage: L(pari('1/poltchebi(5) + O(x^10)'), -10) # Multiply by q^-10
1/5*q^-11 + 4/5*q^-9 + 64/25*q^-7 + 192/25*q^-5 + 2816/125*q^-3 + 8192/125*q^-1 + O(1)
sage: L(pari('O(x^-10)'))
O(q^-10)
"""
from sage.rings.fraction_field_element import is_FractionFieldElement
from sage.rings.polynomial.polynomial_element import is_Polynomial
from sage.rings.polynomial.multi_polynomial_element import is_MPolynomial
if isinstance(x, laurent_series_ring_element.LaurentSeries
) and n == 0 and self is x.parent():
return x # ok, since Laurent series are immutable (no need to make a copy)
elif isinstance(x, pari_gen):
t = x.type()
if t == "t_RFRAC": # Rational function
x = self(self.polynomial_ring()(x.numerator())) / \
self(self.polynomial_ring()(x.denominator()))
return (x << n)
elif t == "t_SER": # Laurent series
n += x._valp()
bigoh = n + x.length()
x = self(self.polynomial_ring()(x.Vec()))
return (x << n).add_bigoh(bigoh)
else: # General case, pretend to be a polynomial
return self(self.polynomial_ring()(x)) << n
elif is_FractionFieldElement(x) and \
(x.base_ring() is self.base_ring() or x.base_ring() == self.base_ring()) and \
(is_Polynomial(x.numerator()) or is_MPolynomial(x.numerator())):
x = self(x.numerator()) / self(x.denominator())
return (x << n)
else:
return laurent_series_ring_element.LaurentSeries(self, x, n)
def _element_constructor_(self, x, n=0):
r"""
Construct a Laurent series from `x`.
INPUT:
- ``x`` -- object that can be converted into a Laurent series
- ``n`` -- (default: 0) multiply the result by `t^n`
EXAMPLES::
sage: R.<u> = LaurentSeriesRing(Qp(5, 10))
sage: S.<t> = LaurentSeriesRing(RationalField())
sage: print R(t + t^2 + O(t^3))
(1 + O(5^10))*u + (1 + O(5^10))*u^2 + O(u^3)
Note that coercing an element into its own parent just produces
that element again (since Laurent series are immutable)::
sage: u is R(u)
True
Rational functions are accepted::
sage: I = sqrt(-1)
sage: K.<I> = QQ[I]
sage: P.<t> = PolynomialRing(K)
sage: L.<u> = LaurentSeriesRing(QQ[I])
sage: L((t*I)/(t^3+I*2*t))
1/2 + 1/4*I*u^2 - 1/8*u^4 - 1/16*I*u^6 + 1/32*u^8 +
1/64*I*u^10 - 1/128*u^12 - 1/256*I*u^14 + 1/512*u^16 +
1/1024*I*u^18 + O(u^20)
::
sage: L(t*I) / L(t^3+I*2*t)
1/2 + 1/4*I*u^2 - 1/8*u^4 - 1/16*I*u^6 + 1/32*u^8 +
1/64*I*u^10 - 1/128*u^12 - 1/256*I*u^14 + 1/512*u^16 +
1/1024*I*u^18 + O(u^20)
TESTS:
When converting from `R((z))` to `R((z))((w))`, the variable
`z` is sent to `z` rather than to `w` (see :trac:`7085`)::
sage: A.<z> = LaurentSeriesRing(QQ)
sage: B.<w> = LaurentSeriesRing(A)
sage: B(z)
z
sage: z/w
z*w^-1
Various conversions from PARI (see also :trac:`2508`)::
sage: L.<q> = LaurentSeriesRing(QQ)
sage: L.set_default_prec(10)
sage: L(pari('1/x'))
q^-1
sage: L(pari('poltchebi(5)'))
5*q - 20*q^3 + 16*q^5
sage: L(pari('poltchebi(5) - 1/x^4'))
-q^-4 + 5*q - 20*q^3 + 16*q^5
sage: L(pari('1/poltchebi(5)'))
1/5*q^-1 + 4/5*q + 64/25*q^3 + 192/25*q^5 + 2816/125*q^7 + O(q^9)
sage: L(pari('poltchebi(5) + O(x^40)'))
5*q - 20*q^3 + 16*q^5 + O(q^40)
sage: L(pari('poltchebi(5) - 1/x^4 + O(x^40)'))
-q^-4 + 5*q - 20*q^3 + 16*q^5 + O(q^40)
sage: L(pari('1/poltchebi(5) + O(x^10)'))
1/5*q^-1 + 4/5*q + 64/25*q^3 + 192/25*q^5 + 2816/125*q^7 + 8192/125*q^9 + O(q^10)
sage: L(pari('1/poltchebi(5) + O(x^10)'), -10) # Multiply by q^-10
1/5*q^-11 + 4/5*q^-9 + 64/25*q^-7 + 192/25*q^-5 + 2816/125*q^-3 + 8192/125*q^-1 + O(1)
sage: L(pari('O(x^-10)'))
O(q^-10)
"""
from sage.rings.fraction_field_element import is_FractionFieldElement
from sage.rings.polynomial.polynomial_element import is_Polynomial
from sage.rings.polynomial.multi_polynomial_element import is_MPolynomial
from sage.structure.element import parent
P = parent(x)
if isinstance(x, self.element_class) and n == 0 and P is self:
return x # ok, since Laurent series are immutable (no need to make a copy)
elif P is self.base_ring():
# Convert x into a power series; if P is itself a Laurent
# series ring A((t)), this prevents the implementation of
# LaurentSeries.__init__() from effectively applying the
# ring homomorphism A((t)) -> A((t))((u)) sending t to u
# instead of the one sending t to t. We cannot easily
# tell LaurentSeries.__init__() to be more strict, because
# A((t)) -> B((u)) is expected to send t to u if A admits
# a coercion to B but A((t)) does not, and this condition
# would be inefficient to check there.
x = self.power_series_ring()(x)
elif isinstance(x, pari_gen):
t = x.type()
if t == "t_RFRAC": # Rational function
x = self(self.polynomial_ring()(x.numerator())) / \
self(self.polynomial_ring()(x.denominator()))
return (x << n)
elif t == "t_SER": # Laurent series
n += x._valp()
bigoh = n + x.length()
x = self(self.polynomial_ring()(x.Vec()))
return (x << n).add_bigoh(bigoh)
else: # General case, pretend to be a polynomial
return self(self.polynomial_ring()(x)) << n
elif is_FractionFieldElement(x) and \
(x.base_ring() is self.base_ring() or x.base_ring() == self.base_ring()) and \
(is_Polynomial(x.numerator()) or is_MPolynomial(x.numerator())):
x = self(x.numerator()) / self(x.denominator())
return (x << n)
return self.element_class(self, x, n)
def __call__(self, x, n=0):
r"""
Coerces the element x into this Laurent series ring.
INPUT:
- ``x`` - the element to coerce
- ``n`` - the result of the coercion will be
multiplied by `t^n` (default: 0)
EXAMPLES::
sage: R.<u> = LaurentSeriesRing(Qp(5, 10))
sage: S.<t> = LaurentSeriesRing(RationalField())
sage: print R(t + t^2 + O(t^3))
(1 + O(5^10))*u + (1 + O(5^10))*u^2 + O(u^3)
Note that coercing an element into its own parent just produces
that element again (since Laurent series are immutable)::
sage: u is R(u)
True
Rational functions are accepted::
sage: I = sqrt(-1)
sage: K.<I> = QQ[I]
sage: P.<t> = PolynomialRing(K)
sage: L.<u> = LaurentSeriesRing(QQ[I])
sage: L((t*I)/(t^3+I*2*t))
1/2 + 1/4*I*u^2 - 1/8*u^4 - 1/16*I*u^6 + 1/32*u^8 +
1/64*I*u^10 - 1/128*u^12 - 1/256*I*u^14 + 1/512*u^16 +
1/1024*I*u^18 + O(u^20)
::
sage: L(t*I) / L(t^3+I*2*t)
1/2 + 1/4*I*u^2 - 1/8*u^4 - 1/16*I*u^6 + 1/32*u^8 +
1/64*I*u^10 - 1/128*u^12 - 1/256*I*u^14 + 1/512*u^16 +
1/1024*I*u^18 + O(u^20)
Various conversions from PARI (see also #2508)::
sage: L.<q> = LaurentSeriesRing(QQ)
sage: L.set_default_prec(10)
sage: L(pari('1/x'))
q^-1
sage: L(pari('poltchebi(5)'))
5*q - 20*q^3 + 16*q^5
sage: L(pari('poltchebi(5) - 1/x^4'))
-q^-4 + 5*q - 20*q^3 + 16*q^5
sage: L(pari('1/poltchebi(5)'))
1/5*q^-1 + 4/5*q + 64/25*q^3 + 192/25*q^5 + 2816/125*q^7 + O(q^9)
sage: L(pari('poltchebi(5) + O(x^40)'))
5*q - 20*q^3 + 16*q^5 + O(q^40)
sage: L(pari('poltchebi(5) - 1/x^4 + O(x^40)'))
-q^-4 + 5*q - 20*q^3 + 16*q^5 + O(q^40)
sage: L(pari('1/poltchebi(5) + O(x^10)'))
1/5*q^-1 + 4/5*q + 64/25*q^3 + 192/25*q^5 + 2816/125*q^7 + 8192/125*q^9 + O(q^10)
sage: L(pari('1/poltchebi(5) + O(x^10)'), -10) # Multiply by q^-10
1/5*q^-11 + 4/5*q^-9 + 64/25*q^-7 + 192/25*q^-5 + 2816/125*q^-3 + 8192/125*q^-1 + O(1)
sage: L(pari('O(x^-10)'))
O(q^-10)
"""
from sage.rings.fraction_field_element import is_FractionFieldElement
from sage.rings.polynomial.polynomial_element import is_Polynomial
from sage.rings.polynomial.multi_polynomial_element import is_MPolynomial
if isinstance(x, laurent_series_ring_element.LaurentSeries) and n==0 and self is x.parent():
return x # ok, since Laurent series are immutable (no need to make a copy)
elif isinstance(x, pari_gen):
t = x.type()
if t == "t_RFRAC": # Rational function
x = self(self.polynomial_ring()(x.numerator())) / \
self(self.polynomial_ring()(x.denominator()))
return (x << n)
elif t == "t_SER": # Laurent series
n += x._valp()
bigoh = n + x.length()
x = self(self.polynomial_ring()(x.Vec()))
return (x << n).add_bigoh(bigoh)
else: # General case, pretend to be a polynomial
return self(self.polynomial_ring()(x)) << n
elif is_FractionFieldElement(x) and \
(x.base_ring() is self.base_ring() or x.base_ring() == self.base_ring()) and \
(is_Polynomial(x.numerator()) or is_MPolynomial(x.numerator())):
x = self(x.numerator())/self(x.denominator())
return (x << n)
else:
return laurent_series_ring_element.LaurentSeries(self, x, n)